Eric Temple Bell on Emmy Noether

“The third great epoch in the extension of arithmetic is that of the twentieth century after 1910. To anticipate, the introduction of general methods into linear algebra, beginning in the first decade of the twentieth century, prepared that vast field of mathematics, first opened up by Hamilton and Grassman in the 1840s, for partial arithmetization in the second and third decades of the century. In 1910, E. Steinitz... proceeding from, and partly generalizing, Kronecker's theory (1881) of "algebraic magnitudes," made a fundamental contribution to the modern theory of (commutative) fields. His work was one of the strongest impulses to the abstract algebra of the 1920s and 1930s, with its accompanying generalized arithmetic. The outstanding figure in the later phase of this development is usually considered to be Emmy Noether... who, with her numerous pupils, laid down the broad foundations of the modern abstract theory of ideals, also a great deal more in the domain of modern algebra. The application of this work to the 'integers' of linear associative algebras affords the ultimate extension up to 1940 of common arithmetic.”

“The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined non-tentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920.”

“The third and last exception to general sterility connects the arithmetic of forms with that other major outgrowth of ancient diophantine analysis, the Gaussian concept of congruence. Dickson in 1907 began the congruencial theory of forms, in which the coefficients of the forms are either natural integers reduced modulo p, p prime, or elements of a Galois field. The linear transformations in the theory, corresponding to those in the classical problem of equivalence, were similarly reduced, and hence modular invariants and covariants were defineable. By 1923 the theory was practically worked out, except for two central difficulties, by Dickson and his pupils. Simplified derivations for some of the results were given (1926) by E. Noether by an application of her methods in abstract algebra.”